Poisson Distribution

Introduction

Assume that an interval is divided into a very large number of equal subintervals so that the probability of the occurrence of an event in any subinterval is very small (e.g., $\le 0.05$). The Poisson distribution models the number of events occurring on that interval, assuming

1. The probability of the occurrence of an event is constant for all subintervals.
2. There can be no more than one occurrence in each subinterval.
3. Occurrences are independent.

直观来说，Poisson Distribution 是二项式分布在 $p\to 0, n\to \infin$ 的情况．当 $np\to \lambda$ 时，二项式分布就会收敛到 Poisson Distribution

$$\mathbb{P}(x|\lambda)=\frac{e^{-\lambda}\lambda^x}{x!}, x=0,1,2,\dots$$

我们用 $X\sim \text{Poisson}(\lambda)$ 来表示．

Mean and Variance

$$\mu_X=\mathbb{E}[X]=\lambda, \sigma^2_X=\text{Var}(X)=\lambda$$